How does SSL encryption happens with default server ssl profile?
This is a really big question, can you perhaps offer a bit more focused question?
Why does the certificate needs to be the same on LTM and pool members?
The certificate configured on an SSL enabled (e.g. has a clientssl profile) does not need to match the certificate configured on the pool members. It's common to have a commercially signed certificate on the VIP and self-signed certificates on pool members.
Can the certificate on client-ssl profile and pool members have the same hostname but different intermediate and root certificate?
I can't see any reason why not. The serverssl profile, which controls negotiation of SSL between the BIG-IP and pool members doesn't need a validated certificate on the pool member, it just needs any certificate in order to be able to negotiate a connection.